The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.
H_{D} | M_{D} | L_{D} | |
---|---|---|---|
H_{G} | 0.40 | 0.48 | 0.12 |
M_{G} | 0.10 | 0.65 | 0.25 |
L_{G} | 0.01 | 0.50 | 0.49 |
Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature $$\left( {{H_G}} \right)$$ then the probability of Delhi also having a high temperature $$\left( {{H_D}} \right)$$ is $$0.40;$$ i.e., $$P\left( {{H_D}|{H_G}} \right) = 0.40.$$ Similarly, the next two entries are $$P\left( {{M_D}|{H_G}} \right) = 0.48$$ and $$P\left( {{L_D}|{H_G}} \right) = 0.12.$$ Similarly for the other rows.
If it is known that $$P\left( {{H_G}} \right) = 0.2,\,\,$$ $$P\left( {{M_G}} \right) = 0.5,\,\,$$ and $$P\left( {{L_G}} \right) = 0.3,\,\,$$ then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______.
$${p_j}\,\, \ge 0,\,j \in \left\{ {0,..........,\,\,\,N} \right\},$$ and $$\,\,\sum\limits_{j = 0}^N {{p_j} = 1,\,\,} $$ define the polynomial function $${g_x}\left( z \right) = \sum\limits_{j = 0}^N {{p_j}{z^j}} .$$ For a certain discrete random variable $$Y$$, there exists a scalar $$\beta $$ $$ \in \left[ {0,1} \right]$$ such that $${g_y}\left( z \right) = {\left\{ {1 - \beta + \left. {\beta z} \right)} \right.^N}.$$ The expectation of $$Y$$ is